Heat a large, shallow 10-inch cast-iron pan or Dutch oven up over medium-high heat.
While the pan heats up, cut the chicken thighs into 1 1/2-inch pieces and season liberally with kosher salt and pepper.
Once the pan is heated, add 2 tablespoons of the ghee and the chicken in a single layer, working in batches and adding more ghee, if needed. Cook for 2 to 3 minutes, flipping the chicken once halfway, until browned evenly. Remove with a slotted spoon or tongs to a plate (the chicken will not be cooked through).
Reduce the heat to medium; add the onion, garlic and sage and cook, stirring, until fragrant and the onions start to soften, 2 to 3 minutes. Add the vermouth, scraping up any browned bits from the bottom of the pan.
Add the chard to the pan and cook, stirring occasionally, until it starts to wilt, 3 to 4 minutes. Next, add the sweet potato and stock. Stir together and season liberally with kosher salt and pepper. Bring to a simmer, then stir in the mascarpone and Parmesan. Return the chicken to the pan, stir then remove from the heat.
Place the phyllo sheets on a baking sheet and cover with a damp towel to prevent them from drying out. Place 1 of the phyllo sheets on a clean work surface then brush or dab with some of the melted butter. Crumple the sheet up like a piece of paper and place on top of the chicken filling. Repeat with the remaining phyllo sheets until the entire surface of the filling is covered, overlapping the sheets, if necessary. Drizzle any remaining butter on top and sprinkle with flaky sea salt.
Transfer the pan to the oven and bake for 10 minutes. Reduce the oven temperature to 375 degrees F and bake until bubbly and the top is golden brown, 20 to 25 minutes. Remove from the oven and let cool for 5 to 10 minutes before serving. If the filling under the crust looks too wet, cook on the stove over medium heat to reduce some of the liquid, 5 to 10 minutes. Serve the pot pie in the pan.